We identify the category of integrable lowest-weight representations of the loop group LGL G of a compact Lie group GG with the linear category of twisted, conjugation-equivariant curved Fredholm complexes on the group GG: namely, the twisted, equivariant matrix factorizations of a super-potential built from the loop rotation action on LGL G. This lifts the isomorphism of K-groups of [FHT1,2, 3] to an equivalence of categories. The construction uses families of Dirac operators.

The FHT-theorem asserts that this construction gives an equivalence of the category of loop group representations at some level τ\tau (Verlinde ring) and the τ\tau-twisted K-theory, equivariant with respect to AdGAd_G, on GG.

In our case, the Verlinde ring is a Frobenius ring / ℤ\mathbb{Z}. It is a quotient of the ring of representations RGR_G by the ideal of characters vanishing at the regular points F=ker(T→k+cT∨)F = ker(T \stackrel{k+c}{\to}T^\vee), where cc is the dual Coxeter number?

Example: What is H0(XΣ\Δ,𝒪(k))H^0(X_{\Sigma \backslash \Delta}, \mathcal{O}(k)) as an LGL G-representation? Turns out the multiplicity of a certain “vacuum” representation inside is equal to dimH0(KG(Σ),𝒪(k))dim H^0(K_G(\Sigma), \mathcal{O}(k))

While complex analysis on such infinite-dimensional manifolds is still out of reach, there exists an algebraic model for XΣ\ΔX_{\Sigma \backslash \Delta} for which we can ask and answer analogous question

For a vector bundle VV on the orbit OλO_\lambda of λ\lambda, which is a sum of line bundles with curvature ωλ\omega_\lambda and which carries a lifted GG-action.

The Dirac index DInd(Oλ,V)D Ind(O_\lambda, V) is a representation of GG and this establishes the Kirillov correspondence

Remarks: For connected GG, VV carries no information beyond its rank

Wehn π1(G)≠0\pi_1(G) \neq 0 the GG-action on VV should be projective and cancel the spinor projective cocycle

The Dirac family consruction (Freed, Hopkins, Teleman) provides an inverse to this, assigning an orbit,…

Input: irreducible representation VλV_\lambda of GG

ω\omega highest weight λ\lambda invariant metric on 𝔤\mathfrak{g}

Uee Kostant’s “cubic Dirac operator” on GG, which is the Dirac operator on GG for the metric connection with canonical torsion coming from H3(G,ℤ)≃ℤH^3(G, \mathbb{Z}) \simeq \mathbb{Z}.

Two key properties of this Dirac operator

[D,ψ(ξ)]=2T(ξ)
[D, \psi(\xi)] = 2 T(\xi)

(translation)

D2=−(λ+p)2
D^2 = - (\lambda + p)^2

on Vλ⊗S±V_\lambda \otimes S^{\pm}

The Dirac family is the ℤ/2\mathbb{Z}/2-graded vector bundle with fiber V⊗S±V \otimes S^{\pm} over 𝔤\mathfrak{g} and odd operator

DξV:=DV+ψ(ξ)
D_\xi^V := D^V + \psi(\xi)

Theorem (FHT): The kernel of DξVD^V_\xi is supported on the orbit of (λ+p)(\lambda + p) and equals (Kirillovlinebundle)⊗(Spinorstonormalbundle)(Kirillov line bundle) \otimes (Spinors to normal bundle) In fact DξVD^V_\xi is a model for the Atiyah-Bott-Shapiro…